What crits more often?

[CTurbo]

Has anyone done the math on how often a reckless attacking Barbarian crits compared to a Champion Fighter?

I'm curious. It's basically...


critting on a 20 with advantage
vs
critting on 18-20 without advantage

[OldTrees1]

18-20 is worth 21/400th more than advantage.

Imagine rolling 2d20 so you have 400 results.
Case A: If you got a 18 or 20, you win. That is 20 cases + 20 cases + 20 cases out of 400 cases. 60/400 = 15%
Cave B: If you roll a 20, you win OR if you roll 1-19 but get a 20 on the 2nd die you win. That is 20 cases + 19 x 1 cases. 39/400 = 9.75%

Explained another way.
A die has a 3/20 chance of getting a 18-20. 3/20=60/400
A die has a 1/20 chance of getting 20.
2 dice have a 1/400 chance of both being 20.
2 dice have a 1/20 chance of getting a 20 on the 1st die + a 1/20 chance of getting a 20 on the 2nd die - the 1/400 chance of both being 20 because we counted it twice already. 1/20+1/20-1/400=39/400

Explained a 3rd way.
A die has a 17/20 chance of not being 18-20.
A die has a 1-(17/20) chance of being a 18-20. 1-(17/20)=3/20=60/400
A die has a 19/20 chance of not being 20.
2 dice have a (19/20)^2 chance of neither being 20.
2 dice have a 1-(19/20)^2 chance of having at least 1 20. 1-(19/20)^2=(400-19^2)/400=39/400


Ranking:
Normal = 20/400
Advantage = 39/400
19-20 = 40/400
18-20 = 60/400
Adv 19-20 =76/400
17-20 = 80/400
16-20 = 100/400
Adv 18-20 = 111/400
15-20 = 120/400
14-20 = 140/400
Adv 17-20 = 144/400

[Desamir]

Advantage roughly doubles your critical chance. If you crit on a 20, advantage is almost equivalent to critting on 19-20. If you crit on 19-20, advantage is almost equivalent to critting on 17-20.

[Christew]

Ten points for thoroughness!

[CTurbo]

18-20 is worth 21/400th more than advantage.

Imagine rolling 2d20 so you have 400 results.
Case A: If you got a 18 or 20, you win. That is 20 cases + 20 cases + 20 cases out of 400 cases. 60/400 = 15%
Cave B: If you roll a 20, you win OR if you roll 1-19 but get a 20 on the 2nd die you win. That is 20 cases + 19 x 1 cases. 39/400 = 9.75%

Explained another way.
A die has a 3/20 chance of getting a 18-20. 3/20=60/400
A die has a 1/20 chance of getting 20.
2 dice have a 1/400 chance of both being 20.
2 dice have a 1/20 chance of getting a 20 on the 1st die + a 1/20 chance of getting a 20 on the 2nd die - the 1/400 chance of both being 20 because we counted it twice already. 1/20+1/20-1/400=39/400

Explained a 3rd way.
A die has a 17/20 chance of not being 18-20.
A die has a 1-(17/20) chance of being a 18-20. 1-(17/20)=3/20=60/400
A die has a 19/20 chance of not being 20.
2 dice have a (19/20)^2 chance of neither being 20.
2 dice have a 1-(19/20)^2 chance of having at least 1 20. 1-(19/20)^2=(400-19^2)/400=39/400


Ranking:
Normal = 20/400
Advantage = 39/400
19-20 = 40/400
18-20 = 60/400
Adv 19-20 =76/400
17-20 = 80/400
16-20 = 100/400
Adv 18-20 = 111/400
15-20 = 120/400
14-20 = 140/400
Adv 17-20 = 144/400

I don't get it. Can you explain further? lol just kidding that was great. Thank you.

Sounds like Champion 18/Barb 2 is the way to go then! hahaha

[Spacehamster]

I don't get it. Can you explain further? lol just kidding that was great. Thank you.

Sounds like Champion 18/Barb 2 is the way to go then! hahaha

Would do 16 champ /4 barb to not miss a single ASI + getting a barb subclass. :) half Orc for bigger crits. :)

[Chronos]

To elaborate on what Desamir said: Advantage means rolling twice as many dice, which means twice as many opportunities to crit, which (approximately) means critting twice as often. But it's not quite twice as often, because occasionally when you roll with advantage, both of the dice will come up natural 20s, but when that happens, you still only get one crit. In other words, occasionally you'll have a "wasted" crit.

Comparing advantage with a 19-20 range to a hypothetical 17-20 crit range, the same effect shows up, but to a greater degree, because with a crit on a 19 or 20, it'll be more common (though still pretty rare) to get a double crit (and hence waste one of the rolls).

On the other hand, if you somehow had an 11-20 crit range (10 numbers) with advantage, that would be very different from somehow getting a 1-20 crit range (20 numbers), because now double crits that waste good rolls are going to be quite common.

[Man_Over_Game]

Y'all are working too damn hard on this.

https://anydice.com/program/1c7a7

Make sure the left is Table, the right is At Least, compare Advantage @20 vs. 1d20@18,19.

I will say that, in terms of damage value, the champion's first bonus to crit (5%) by itself is worth less than just a +1 to hit, mostly because 5% of 2d6 (7 average) is still crap.

Advantage and a crit bonus do work well together, but you'll have much better luck combining Hexblade, Elven Accuracy and Vengeance Paladin (27% crit chance by level 5, one target per Short Rest, bumps to 47% per turn on level 6) than you would across an almost 20 level build with Barbarian and Champion. Especially once you throw Divine Smite into the mix.

[Bobthewizard]

Has anyone done the math on how often a reckless attacking Barbarian crits compared to a Champion Fighter?

I'm curious. It's basically...


critting on a 20 with advantage
vs
critting on 18-20 without advantage

Chances of getting a crit on a 20 with advantage is 9.75%

Chances of getting a crit on 18-20 without advantage is 15%, so this wins.

However, advantage gives you a better chance to hit otherwise, so DPR may be better with advantage depending on AC and to-hit bonus.

[Snownine]

18-20 is worth 21/400th more than advantage.

Imagine rolling 2d20 so you have 400 results.
Case A: If you got a 18 or 20, you win. That is 20 cases + 20 cases + 20 cases out of 400 cases. 60/400 = 15%
Cave B: If you roll a 20, you win OR if you roll 1-19 but get a 20 on the 2nd die you win. That is 20 cases + 19 x 1 cases. 39/400 = 9.75%

Explained another way.
A die has a 3/20 chance of getting a 18-20. 3/20=60/400
A die has a 1/20 chance of getting 20.
2 dice have a 1/400 chance of both being 20.
2 dice have a 1/20 chance of getting a 20 on the 1st die + a 1/20 chance of getting a 20 on the 2nd die - the 1/400 chance of both being 20 because we counted it twice already. 1/20+1/20-1/400=39/400

Explained a 3rd way.
A die has a 17/20 chance of not being 18-20.
A die has a 1-(17/20) chance of being a 18-20. 1-(17/20)=3/20=60/400
A die has a 19/20 chance of not being 20.
2 dice have a (19/20)^2 chance of neither being 20.
2 dice have a 1-(19/20)^2 chance of having at least 1 20. 1-(19/20)^2=(400-19^2)/400=39/400


Ranking:
Normal = 20/400
Advantage = 39/400
19-20 = 40/400
18-20 = 60/400
Adv 19-20 =76/400
17-20 = 80/400
16-20 = 100/400
Adv 18-20 = 111/400
15-20 = 120/400
14-20 = 140/400
Adv 17-20 = 144/400

Now that is a breakdown.

[McSkrag]

18-20 is worth 21/400th more than advantage.

Imagine rolling 2d20 so you have 400 results.
Case A: If you got a 18 or 20, you win. That is 20 cases + 20 cases + 20 cases out of 400 cases. 60/400 = 15%
Cave B: If you roll a 20, you win OR if you roll 1-19 but get a 20 on the 2nd die you win. That is 20 cases + 19 x 1 cases. 39/400 = 9.75%

Explained another way.
A die has a 3/20 chance of getting a 18-20. 3/20=60/400
A die has a 1/20 chance of getting 20.
2 dice have a 1/400 chance of both being 20.
2 dice have a 1/20 chance of getting a 20 on the 1st die + a 1/20 chance of getting a 20 on the 2nd die - the 1/400 chance of both being 20 because we counted it twice already. 1/20+1/20-1/400=39/400

Explained a 3rd way.
A die has a 17/20 chance of not being 18-20.
A die has a 1-(17/20) chance of being a 18-20. 1-(17/20)=3/20=60/400
A die has a 19/20 chance of not being 20.
2 dice have a (19/20)^2 chance of neither being 20.
2 dice have a 1-(19/20)^2 chance of having at least 1 20. 1-(19/20)^2=(400-19^2)/400=39/400


Ranking:
Normal = 20/400
Advantage = 39/400
19-20 = 40/400
18-20 = 60/400
Adv 19-20 =76/400
17-20 = 80/400
16-20 = 100/400
Adv 18-20 = 111/400
15-20 = 120/400
14-20 = 140/400
Adv 17-20 = 144/400

Awesome mathing! Seriously, thanks for the detailed breakdown.

How would elven accuracy affect the odds? And what about 2nd breakfeast?

[OldTrees1]

Awesome mathing! Seriously, thanks for the detailed breakdown.

How would elven accuracy affect the odds? And what about 2nd breakfeast?

Elven accuracy means 3 dice, so let's talk about the 8000 cases.

When I hit 3+ independant events (3 dice) I like to think in terms of "Not"s. One or more successes is not failing every time and failing one time is not succeeding one time.

at_least_one_success = 1 - (all_failures) = 1 - failure_in_one_try^number_of_tries
= 1 - (1 - success_in_one_try) ^ number_of_tries

Elven Accuracy = 1 - ( 1 - 1/20) ^ 3 = 1 - ( 19/20 ) ^ 3 = (8000 - 19^3) / 8000 = 1141/8000
Elven Accuracy 19-20 = 1 - ( 1 - 2/20) ^ 3 = 1 - ( 18/20 ) ^ 3 = (8000 - 18^3) / 8000 = 2168/8000
Elven Accuracy 18-20 = 1 - ( 1 - 3/20) ^ 3 = 1 - ( 17/20 ) ^ 3 = (8000 - 17^3) / 8000 = 3087/8000

Ranking:
Normal = 400/8000
Advantage = 780/8000
19-20 = 800/8000
Elven = 1141/8000
18-20 = 1200/8000
Adv 19-20 =1520/8000
17-20 = 1600/8000
16-20 = 2000/8000
Elven 19-20 = 2168/8000
Adv 18-20 = 2220/8000
15-20 = 2400/8000
14-20 = 2800/8000
Adv 17-20 = 2880/8000
Elven 18-20 = 3087/8000
13-20 = 3200/8000

But there are other ways to think about it.
Of the 8000 cases, 400 of them correspond to rolling a 20 on the first die. Same goes for the 2nd and 3rd dice. However those 1200 cases are double or even triple counting some cases. There are 20 cases where the 1st and 2nd die got 20s. Same with 1st & 3rd and 2nd & 3rd. We are double counting those cases so we can subtract them. However there is 1 case where all 3 dice got a 20. We counted it 3 times and subtracted it 3 times. So we add it back in one last time. 400x3-20x3+1=1141 cases out of 8000.

Or
There is 1 case where there are exactly 3 20s. There are 3x 3x 19 cases where there are exactly 2 20s. There are 3x 19^2 cases where there is exactly 1 20. 1 + 3x19 + 3x19^2 = 1141 out of 8000.

2nd breakfast means toast with an omelet you made while eating the 1st breakfast. Maybe with some oatmeal.

[AttilatheYeon]

Would do 16 champ /4 barb to not miss a single ASI + getting a barb subclass. :) half Orc for bigger crits. :)

18th level is when Champion gets crit 18-20. So for this build the last ASI is worth less. Also, 6 ASIs are probably plenty for most SAD builds.

[MaxWilson]

I don't get it. Can you explain further? lol just kidding that was great. Thank you.

Sounds like Champion 18/Barb 2 is the way to go then! hahaha

It's better to just spend one of your Fighter attacks to Shove enemies prone and get advantage that way. You don't miss out on your fourth attack, and shoving an enemy prone helps melee-oriented allies as well yourself, imposes disadvantage on that enemy's opportunity attacks until it has a chance to stand up, and costs it half its movement next turn.

With Polearm Master and Prodigy (Athletics), you can Shove one enemy prone reliably and then make four attacks per turn. Limitation: doesn't work on Huge monsters unless you've been Enlarged, and doesn't work on Gargantuan monsters at all. But it's still better than Recklessly giving all of your enemies advantage, IMO.

[x3n0n]

18th level is when Champion gets crit 18-20. So for this build the last ASI is worth less. Also, 6 ASIs are probably plenty for most SAD builds.

FWIW, I think this is at 15th:

Superior Critical
Starting at 15th level, your weapon attacks score a critical hit on a roll of 18–20.

That said, 18th seems like it'd be juicy for anyone doing Reckless Attack turn after turn:

Survivor
At 18th level, you attain the pinnacle of resilience in battle. At the start of each of your turns, you regain hit points equal to 5 + your Constitution modifier if you have no more than half of your hit points left. You don’t gain this benefit if you have 0 hit points.

:)

It's better to just spend one of your Fighter attacks to Shove enemies prone and get advantage that way. You don't miss out on your fourth attack, and shoving an enemy prone helps melee-oriented allies as well yourself, imposes disadvantage on that enemy's opportunity attacks until it has a chance to stand up, and costs it half its movement next turn.

With Polearm Master and Prodigy (Athletics), you can Shove one enemy prone reliably and then make four attacks per turn. Limitation: doesn't work on Huge monsters unless you've been, and doesn't work on Gargantuan monsters at all. But it's still better than Recklessly giving all of your enemies advantage, IMO.

I read that several times and was confused because I couldn't figure out what Polearm Master had to do with it, but I think I have it now:

Replace your first attack in the Attack action with a Shove-prone (which will often work because of "Expertise" in Athletics).
Take the rest of your attacks with a Halberd or Glaive.
Take your bonus action d4 attack with the butt-end.

Since you have so many attacks, it's probably even worthwhile to try again for a second Shove if the first one fails.

That said, I've also seen the argument that says you WANT to give them incentive them to attack you instead of the squishies (assuming there are some in your party).

[MaxWilson]

I read that several times and was confused because I couldn't figure out what Polearm Master had to do with it, but I think I have it now:

Replace your first attack in the Attack action with a Shove-prone (which will often work because of "Expertise" in Athletics).
Take the rest of your attacks with a Halberd or Glaive.
Take your bonus action d4 attack with the butt-end.

Since you have so many attacks, it's probably even worthwhile to try again for a second Shove if the first one fails.

Yes, that is what I meant. The tradeoffs are not simple but overall I feel that the full-Fighter approach comes out ahead by endgame, especially if there are other melee fighters in the party (including summoned creatures). Barb 2/Fighter pulls temporarily ahead at levels 13-17 due to those being mostly dead levels for Champion, but even then I'd use Reckless sparingly.

That said, I've also seen the argument that says you WANT to give them incentive them to attack you instead of the squishies (assuming there are some in your party).

The only way to do that is to make yourself squishier than the squishies, at least in the monsters' eyes. Much depends upon whether the DM rules that reduced damage from resistance or immunity looks like full damage to the attacker--but keeping the reduction hidden is arguably worse for the PCs than for the monsters, despite fringe benefits to the Barbarian (looking squishier than he really is).

P.S. If you want to mess with monsters' targeting, having the wizard cast Disguise Self on herself or Seeming on everybody is lovely. IMO. Make squishies look like tanks and tanks (and zombies) look like glass cannons.

[CTurbo]

The Fighter 20 vs Fighter 18/Barb 2 question is an interesting one, and one that we're not all going to agree with.

I think trading a 4th attack and 1 ASI for Unarmored Defense, Danger Sense, and 2 Rages per day is not terrible. Especially considering I'd take the 2 levels of Barb first. Then Survivor becomes a decent capstone, and you still get 6 ASIs.

I do think Champion 3/Barbarian 17 makes a better crit fishing build, it's just not as interesting.

[Chronos]

Quoth Man Over Game:

I will say that, in terms of damage value, the champion's first bonus to crit (5%) by itself is worth less than just a +1 to hit, mostly because 5% of 2d6 (7 average) is still crap.

To explain, the expanded crit range means that, on one attack roll out of 20, you'll be turning a regular hit into a crit, and thereby gaining weapon dice worth of damage. But an extra +1 to hit means that, on one attack roll out of 20, you'll be turning a miss into a hit, and thereby gaining weapon dice plus ability modifier (and other static bonuses, like from a magic weapon or rage) worth of damage. The crit range might feel like more, but that's only because you know when it's working: When the die shows a 19, you know that your crit range helped you, but the DM probably doesn't tell you that your attack just barely hit, meaning the +1 helped you.

[ezekielraiden]

To explain, the expanded crit range means that, on one attack roll out of 20, you'll be turning a regular hit into a crit, and thereby gaining weapon dice worth of damage. But an extra +1 to hit means that, on one attack roll out of 20, you'll be turning a miss into a hit, and thereby gaining weapon dice plus ability modifier (and other static bonuses, like from a magic weapon or rage) worth of damage. The crit range might feel like more, but that's only because you know when it's working: When the die shows a 19, you know that your crit range helped you, but the DM probably doesn't tell you that your attack just barely hit, meaning the +1 helped you.

Correct. Heck, even 10% chance of rolling extra weapon dice isn't very good: average of 2d6 is 7, so you're getting (on average) 0.7 extra damage per swing, while +1 means an extra 0.05*(7+N) where N is your ability modifier, magic weapon stuff, and whatever else that triggers on hits. If N is 7 or more--not super hard with all the various sources of bonus damage out there--then that +1 is literally just as valuable as, or more valuable than, the "special" extra-crits benefit of the Champion.

In fact, a Paladin (for example) gets Improved Divine Smite at level 11. That's +1d8, or ~4.5 on average per hit. Assuming a reasonable +4 ability modifier at that level, the Paladin gets more out of that +1 to hit than the Champion gets from its level 15 feature!

[Man_Over_Game]

But an extra +1 to hit means that, on one attack roll out of 20, you'll be turning a miss into a hit, and thereby gaining weapon dice plus ability modifier (and other static bonuses, like from a magic weapon or rage) worth of damage.

Well, it's worse than that.

On average, a character has about a 65% chance to hit, or a 35% chance to miss. Reduce that 35% to 30%, and that's about a 14% difference in your chance to hit. That is, a single +1 to hit roughly increases your overall damage by 14%. A crit is only 100% of your weapon damage 5% of the time (as it doesn't matter how easy the target it is to hit) which on a good scenario is a 2d6 (for +.35 damage).

Thing is, you roughly deal 10 damage per hit. That means a 5% crit boost only increases your total damage by about 3.5%



+1 to hit = 14% increased damage.
+5% to crit = 3.5% increased damage.

Champion's pretty bad, yo. Heck, assuming I did my math right, you could allow the Champion to crit on a 17 or higher at level 3 (roughly a flat 15% crit chance increase) and it'd still be roughly worse than a +1 to hit (until you get more dice damage than 2d6).

[OldTrees1]

Well, it's worse than that.

On average, a character has about a 65% chance to hit, or a 35% chance to miss. Reduce that 35% to 30%, and that's about a 14% difference in your chance to hit. That is, a single +1 to hit roughly increases your overall damage by 14%. A crit only accounts for 5% of your weapon damage, as it requires the result on the die to be a certain amount regardless of the target's AC.

Correction:
If you hit 65% (aka 13/20) of the time, and you crit on 20s, then you crit on 1/13 of your hits.
1 x crit bonus damage / (1 x crit bonus damage + 13 x normal damage) = fraction of damage due to crits.
Increasing the crit range by 1 increases it to 2/13 of your hits.
2 x crit bonus damage / (2 x crit bonus damage + 13 x normal damage) = the new fraction of damage due to crits.
1 x crit bonus damage / (1 x crit bonus damage + 13 x normal damage) = the fraction of damage increase from the expanded crit range.

I am going to presume this is a 2d6+6 attack (+1 Greatsword, Str 20, no fighting style) to fill in the math.

+1 crit range: 2d6 / (2d6 + 13 x 2d6 + 13 x 6) ~ 7 / (7 + 13 x 13) = 7 / 176 = 3.9772%
+1 attack: 2d6+6 / (2d6 + 13 x 2d6 + 13 x 6) ~ 14 / 176 ~ 7.954%

This is interesting, let's generalize this.

Increasing your proc chance (p/20) by 1 increases you instances by
Normal: 1 / 20
Adv: (2p+1) / 400
Elven: (3p^2+3p+1) / 8000

Obviously +1 is always better since it increases 2 sources of damage and its proc chance was greater (which matters for adv and Elven where it scales based on prior proc chance). But how much better is trickier math.

[ezekielraiden]

Well, it's worse than that.

On average, a character has about a 65% chance to hit, or a 35% chance to miss. Reduce that 35% to 30%, and that's about a 14% difference in your chance to hit. That is, a single +1 to hit roughly increases your overall damage by 14%. A crit is only 100% of your weapon damage 5% of the time (as it doesn't matter how easy the target it is to hit) which on a good scenario is a 2d6 (for +.35 damage).

Thing is, you roughly deal 10 damage per hit. That means a 5% crit boost only increases your total damage by about 3.5%



+1 to hit = 14% increased damage.
+5% to crit = 3.5% increased damage.

Champion's pretty bad, yo. Heck, assuming I did my math right, you could allow the Champion to crit on a 17 or higher at level 3 (roughly a flat 15% crit chance increase) and it'd still be roughly worse than a +1 to hit (until you get more dice damage than 2d6).

I'm fairly sure that "relative increase in chance to hit" is not directly comparable, to crits, and it certainly doesn't mean 14% increased damage. Getting +1 to hit means 5% of full base attack damage--because, on average, you get (hit probability)*(average weapon die value + static bonuses) damage, not (relative increase in chance to hit)*(avg dmg + static bonuses). Critting gives you only the dice you roll when you crit, which are usually inferior to your total static mods.

If we were comparing as close as apples to apples as we can get here, we'd be comparing relative chance to hit increase (14%) to relative increase in percentage of hits that are crits. And that relative increase is 100%, because you crit on twice as many results of the d20 as you did before (single vs two)--10% is 100% bigger than 5%. Relative increase multiplied by damage-per-hit or damage-per-crit doesn't actually give you the expected value for damage--that calculation only works when it's the probability of an event (like hitting) times the numerical weight of that event (like the average die roll + static mods).

[Chronos]

Assertion: For a character with no static damage bonus (i.e., no ability score modifier, rage, etc.), a +1 to hit will result in exactly the same average damage as an expansion of the crit range by 1 point, no matter what the attack bonus or AC (provided that we're not in hit-only-on-a-crit or miss-only-on-a-1 territory). The benefit of the +1 over the expanded crit range is entirely due to the static damage bonus.

Do you dispute that, Man Over Game?

[MaxWilson]

Champion's pretty bad, yo. Heck, assuming I did my math right, you could allow the Champion to crit on a 17 or higher at level 3 (roughly a flat 15% crit chance increase) and it'd still be roughly worse than a +1 to hit (until you get more dice damage than 2d6).

I tried messing around with the Champion's expanded crit range, but eventually landed on a different solution: when a Champion crits, instead of rolling double weapon dice, he doubles all damage including static modifiers.

The reason I prefer this solution is that people already overestimate how impactful Improved Critical is, and expanding the crit range to 17+ makes them think it's overpowered whereas it's actually only average, whereas making Improved Critical double static modifiers just aligns Improved Critical with how powerful people intuitively believe it already is.

Instead of being total garbage, it is about as powerful as +1 to hit: somewhat less powerful than +2 to your attack stat (unless you have additional synergies) but somewhat more powerful offensively than halfling luck.

[OldTrees1]

Assertion: For a character with no static damage bonus (i.e., no ability score modifier, rage, etc.), a +1 to hit will result in exactly the same average damage as an expansion of the crit range by 1 point, no matter what the attack bonus or AC (provided that we're not in hit-only-on-a-crit or miss-only-on-a-1 territory). The benefit of the +1 over the expanded crit range is entirely due to the static damage bonus.

Do you dispute that, Man Over Game?

That assumes and is accurate for a normal attack rather than advantage or elven accuracy.

Let's say you hit 11-20 and crit 18-20 but are attacking with advantage.
You hit 300/400 and crit 111/400.
+1 attack increases the hit to 319/400. An increase of 19/400.
+1 crit increases the crit to 144/400. An increase of 33/400.
Since the crit adds the same damage as an extra hit (due to no static bonus), the +1 crit is stronger than +1 attack in this example.
I am curious, if this were a greatsword (2d6~7), how much static bonus would break even in this example.
7 * 33/400 = 7 * 19/400 + X * 19/400
7 * 14 = X * 19
7 * 14 / 19 = X
So at advantage, with a hit of 11-20 and a crit of 18-20, swinging with a 2d6+5.15789474, the warrior is ambivalent over +1 attack vs +1 crit.
Double checking my math:
+1 attack (10-20) with crit 18-20 and 2d6+5 has expected damage
(319/400) * (2d6+5) + (111/400) * (2d6) = 11.5125
attack 10-20 with +1 crit (17-20) and 2d6+5 has expected damage
(300/400) * (2d6+5) + (144/400) * (2d6) = 11.52
+1 attack (10-20) with crit 18-20 and 2d6+6 has expected damage
(319/400) * (2d6+6) + (111/400) * (2d6) = 12.31
attack 10-20 with +1 crit (17-20) and 2d6+6 has expected damage
(300/400) * (2d6+6) + (144/400) * (2d6) = 12.27

[Man_Over_Game]

Assertion: For a character with no static damage bonus (i.e., no ability score modifier, rage, etc.), a +1 to hit will result in exactly the same average damage as an expansion of the crit range by 1 point, no matter what the attack bonus or AC (provided that we're not in hit-only-on-a-crit or miss-only-on-a-1 territory). The benefit of the +1 over the expanded crit range is entirely due to the static damage bonus.

Do you dispute that, Man Over Game?

I do.

This is because there are more on-hit benefits than their are strictly damage-dice benefits, as the damage-dice benefits are ALSO an on-hit benefit.

So while Hunter's Mark is both, something like an attack with Sentinel is not. There are fewer crit-specific benefits (orcs and barbarians are it, I think) than there are on-hit benefits that do more than deal damage. Magic weapons have a few crit-specific benefits, but I still think they'd fall short to some feat/class abilities.

[Man on Fire]

Two weapon fughting 15th level champion has around 97% chance to crit at least once

[SLOTHRPG95]

2nd breakfast means toast with an omelet you made while eating the 1st breakfast. Maybe with some oatmeal.

Correction: while not super-useful for crit-fishing, second breakfast still has an impact on the probability of a crit. Updated ranking is below.

Ranking:
Normal = 400/8000
Second Breakfast = 420/8000
Advantage = 780/8000
19-20 = 800/8000
Second Breakfast w/ Advantage = 817.95/8000
S.B. 19-20 = 840/8000
Elven = 1141/8000
18-20 = 1200/8000
S.B. 18-20 = 1260/8000
Adv 19-20 =1520/8000
S.B. Adv 19-20 = 1591.8/8000
17-20 = 1600/8000
16-20 = 2000/8000
Elven 19-20 = 2168/8000
Adv 18-20 = 2220/8000
S.B. Adv 18-20 = 2321.55/8000
15-20 = 2400/8000
14-20 = 2800/8000
Adv 17-20 = 2880/8000
Elven 18-20 = 3087/8000
13-20 = 3200/8000

[OldTrees1]

Correction: while not super-useful for crit-fishing, second breakfast still has an impact on the probability of a crit. Updated ranking is below.

Ah, right. Thanks for catching that math error.
It really is a shame Elves have not hear of elevenses. Otherwise we could have Elven Elevenses.

[Chronos]

Quoth Man_Over_Game:

I do.

This is because there are more on-hit benefits than their are strictly damage-dice benefits, as the damage-dice benefits are ALSO an on-hit benefit.

Good point; I should have specified "for purposes of average damage". Another difference between expanded crit range and +1 to hit is that the expanded crit range increases variance, which is usually to the players' detriment.